\section{Introduction}
Maxwell's equations are simply the mathematical relationships of
the electric and magnetic fields with respect to their charge and
current sources \cite{Iskander:1992}. There are four equations
that make up Maxwell's equations, namely:
\begin{itemize}
    \item Gauss's Electric Field Law
    \item Gauss's Magnetic Field Law
    \item Faraday's Law
    \item Ampere's Law
\end{itemize}
Each equation was named after it's major contributing
investigator. For the most part, these equations were developed by
quantifying experimental results. In 1873, Maxwell introduced a
displacement current into Ampere's law which predicted
electromagnetic wave propagation \cite{Maxwell:1873}. Maxwell's
contribution unified the study of electronics and magnetics into
one common field, electromagnetics. For this reason, the four
equations are called Maxwell's equations. Maxwell's prediction of
electromagnetic wave propagation, based on mathematical and
theoretical considerations, was later verified experimentally be
Hertz and Marconi \cite{Pozar:1998}.

Maxwell's equations, proved and tested over time, are usually
considered the fundamental starting point in solving an
electromagnetic problem, and therefore will be the starting point
here. Covered in this chapter will be many important equations and
theorems used as the building blocks for the mode matching theory. 
The reader is assumed to have a basic background in
electromagnetic theory. A number of basic and advanced books on
electromagnetic theory is included in the bibliography for
reference\cite{Balanis:1989,Collin:1992,Harrington:1961,Iskander:1992}.
\section{Maxwell's Equations in Differential Form}
\input{./src/MaxwellDiffLossy.tex}
\section{Maxwell's Equations in Integral Form}
\input{./src/MaxwellIntLossy.tex}
\section{Constitutive Parameters}
\input{./src/ConstitutiveParameters.tex}
\section{Fields in Media}
\section{Boundary Conditions}
\input{./src/BoundaryConditions.tex}
\subsection{General Material Interface}
\subsection{Dielectric--Perfect Electric Conductor Interface}
\subsection{Dielectric--Perfect Magnetic Conductor Interface}

\section{Wave Equation}
\input{./src/WaveEqnDerivation.tex}

\section{Power and Energy}\label{sec:PowerandEnergy}
\input{./src/PowerandEnergy.tex}

\subsection{Time Domain}
\subsection{Frequency Domain}

\section{Theorems and Identities}
\subsection{Stokes Theorem}
\subsection{Divergence Theorem}
\subsection{Green's Identities}
\subsection{Reaction Theorem}
